Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36009 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,28)(19,24,32,40)(20,25,35,43)(21,26,22,27)(29,37,46,48)(30,38,31,39)(33,41,45,47)(34,42,36,44);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,40)(20,33,35,45)(21,30,22,31)(25,41,43,47)(26,38,27,39)(29,36,46,34)(37,44,48,42);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,34)(22,36)(23,35)(24,37)(26,42)(27,44)(28,43)(30,45)(31,33)(32,46)(38,47)(39,41)(40,48);(2,6)(5,7)(17,21,23,22)(18,26,28,27)(19,30,32,31)(20,34,35,36)(24,38,40,39)(25,42,43,44)(29,45,46,33)(37,47,48,41);(3,8)(5,7)(19,31)(21,22)(24,39)(26,27)(29,33)(30,32)(34,36)(37,41)(38,40)(42,44)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6597659);